3
$\begingroup$

Find the pattern in the sequence:

10 11 21 247 29 391 ? ? ?

$\endgroup$
7
$\begingroup$

The next three numbers are:

1271, 47, 1591

The pattern:

Let $a_i$ denote the $i$-th number in the sequence, $p_i$ denote the $i$-th prime number.

We have
$a_1 = 10 = 2 \times 5 = p_1 \times p_3 $
$a_2 = 11 = p_5 $
$a_3 = 21 = 3 \times 7 = p_2 \times p_4 $
$a_4 = 247 = 13 \times 19 = p_6 \times p_8 $
$a_5 = 29 = p_{10} $
$a_6 = 391 = 17 \times 23 = p_7 \times p_9 $

Therefore for $i \ge 0$:
$a_{3i+1} = p_{5i+1} \times p_{5i+3}$
$a_{3i+2} = p_{5i+5}$
$a_{3i+3} = p_{5i+2} \times p_{5i+4}$

Plug $2$ into $i$ we can get:
$a_7 = p_{11} \times p_{13} = 31 \times 41 = 1271$
$a_8 = p_{15} = 47$
$a_9 = p_{12} \times p_{14} = 37 \times 43 = 1591$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.